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\begin{document}

\title[Galois cohomology]{The image of the map from group cohomology to Galois cohomology}
 
\author{M.Tezuka and N.Yagita}
 


\address{ Department of mathematics, Faculty of Science, Ryukyu University, Okinawa, Japan\\
Department of Mathematics, Faculty of Education, 
Ibaraki University,
Mito, Ibaraki, Japan}
 
\email{ tez@sci.u-ryukyu.ac.jp,  yagita@mx.ibaraki.ac.jp, }
\keywords{cohomolgy invariant, classifying spaces, motivic cohomology}
\subjclass{Primary 11E72, 12G05; Secondary 55R35}

\begin{abstract}
We study the image of the natural map from group cohomology to Galois cohomology  by using motivic
cohomology of classifying spaces.
\end{abstract}


\maketitle


\section{Introduction}


Let $k$ be a field of $ch(k)=0$, which contains a $p$-th root of unity.
Let $G$ be a split affine algebraic group over $k$
and $W$ a faithful representation of $G$.
Then $G$ acts also on the function field $k(W)$.
Let $k(W)^G$ be the invariant filed.  Then we have the natural
quotient map of groups
$ q: Gal(\bar{k}{(}\bar{W}{)}/k(W)^G)\to G.$
This induces the map of cohomologies
\[q^*: H^*(G;\bZ/p)\to H^*(k(W)^G;\bZ/p).\]

The purpose of this paper is to study the image $Im(q^*)$
 by using the motivic cohomology defined by
Suslin and Voevodsky [Vo1,3].
The image $Im(q^*)$ is called the $stable$ cohomology
in [Bo], [Bo-Pe-Ts].  The kernel $Ker(q^*)=Ng$ is called the (geometric)
 negligible ideal [Pe],[Sa].

Let $H^{*,*'}(X;\bZ/p)$ be the $mod(p)$ motivic cohomology.
Let $0\not =\tau\in H^{0,1}(Spec(k);\bZ/p)\cong \bZ/p$.
Using affirmative solution of the Bloch-Kato conjecture by Voevodsky
(and hence Beilinson-Lichtenbaum conjecture), the map $q^*$ is decomposed as
\[ q^*:H^{*,*}(BG;\bZ/p)\to H^{*,*}(BG;\bZ/p)/(\tau) \to H^{*,*}(Spec(k(W)^G);\bZ/p).\]
where $H^{*,*}(BG;\bZ/p)/(\tau)=H^{*,*}(BG;\bZ/p)/(\tau H^{*,*-1}(BG;\bZ/p))$
and $BG$ is the classifying space of $G$ defined by Totaro and Voevodsky ([To], [Vo1,4]).

By the Belinson-Lichtenbaum conjecture and the work of Bloch-Ogus [Bl-Og], we know
\[ H^{*,*}(BG;\bZ/p)/(\tau)\subset H^0_{Zar}(BG;H^*_{\bZ/p})\subset H^*(k(W)^G;\bZ/p).\]
Here $H^*_{\bZ/p}$ is the Zarisky sheaf induced from the presheaf
$H_{et}^*(V;\bZ/p)$ for open subset $V$ of $BG$.  Therefore we see ([Or-Vi-Vo]) 
\[ Im(q^*)=H^*(BG;\bZ/p)/(Ng)\cong H^{*,*}(BG;\bZ/p)/(\tau).\]
Note that the right hand side ring does not depend on the choice of $W$.
We also note that the ideal $(Ng)$ coincides the coniveau filtration $N^1H^*(BG;\bZ/p)$
defined by Grothendieck.  

In this paper we compute $Im(q^*)$ when  $k=\bC$ for abelian $p$-groups,
symmetric group $S_n$, $O_n$, $SO_n$, $Spin_n$, $PGL_p$ and exceptional groups.
Extra special $p$-groups are also studied.  
For example, we see
$H^*(BSpin_n;\bZ/2)/(Ng)\cong \bZ/2$ for all $n\ge 6$.

Recall that the cohomology invariant $Inv^*(G;\bZ/p)$ is a ring of natural maps
$H^1(F;G)\to H^*(F;\bZ/p)$ for finitely generated fields $F$ over $k$.
This ring  is very well studied for example see [Ga-Me-Se].  
In particular, it is very useful to compute the essential dimension $ed(G)$ of $G$
([Re], [Br-Re-Vi]).
Moreover, Totaro proved that
\[ Inv^*(G;\bZ/p)\cong H^0_{Zar}(BG;H_{\bZ/p}^*)\]
in a letter to Serre [Ga-Me-Se].
Hence $Im(q^*)\subset Inv^*(G;\bZ/p)$.
 We use these results for some parts of this paper,
however we also give new explanations of $Inv^*(G;\bZ/p)$ for the case $k=\bC$.
For example,the image of (topological) Stiefel-Whitney class $w_i$ of the map
\[H^*(BO_n;\bZ/2)\to H^*(BO_n;\bZ/2)/(Ng)\subset Inv^*(O_n;\bZ/2)\]
is indeed the Stiefel-Whitney class $w_i$ defined by Milnor and Serre
as the natural function from quadratic forms to Milnor $K$-theories. 


All examples stated above are detected by abelian $p$-subgroups $A$ 
of $G$, i.e., the restriction map
\[Res: H^*(BG;\bZ/p)/(Ng)\to \Pi _{A}H^*(BA;\bZ/p)/(Ng)\]
is injective. (Indeed, most of the above cases are  detected by only one elemenary abelian $p$-subgroup.)

Of course this detected property does not hold for general $G$.
However to give examples is not so easy.
Indeed, for a $p$-group $G$ of exponent $p$, if $H^2(BG;\bZ/p)/(Ng)$ is not detected by any $A\cong \bZ/p\oplus\bZ/p$,
then $G$ is a counter example of the Noether's problem, namely,
$k(W)^G$ is not purely transcendental over $k$.
The examples of  Saltman and Bogomolov are essentially of these types [Sa].

For each $n>1$, we give an example $G_n$ of a $p$-group $p\ge 3$, such that $H^{2n}(G_n;\bZ/p)$ is  not detected
by abelian $p$-subgroups, while it does not implies a counter example
of Noether's problem.
Here the composition $Q_{2n-2}...Q_0$ of Milnor operations  is used to see 
$x\not \in Ng$ given $x\in H^{2n}(BG;\bZ/p)$. 


The second author learned theories of cohomology invariants 
from the discussions with Burt Totaro and Patrick Brosnan.
Discussions with Alexander Vishik, Zinovy Reichstein, Kirill Zainoulline,
Nikita Semenov, Masaharu Kaneda, Charles Vial and Dinesh Deshpande
are very helpful to write this paper. The authors thank them very much.



 



%Let $G$ be a finite group and $W$ be a faithful representation
%of $G$ over $k$.
%Then $G$ acts also on the function field $k(W)$.
%Noether's problem is to determine whether the invariant field $k(W)^G$
%is purely transcendental over $k$.

%Using unramified cohomology  $H_{nr}^2(k(W)^G;\bQ/\bZ)$,
%D.Saltman gave the first examples of $G$ such that
%$k(W)^G$ is not purely transcendental.
%F. Bogomolov gives a formula which expresses this cohomology group
%in terms of cohomology of the group $G$.
%E.Peyre worked on $H^3_{nr}(-;\bQ/\bZ)$ and gets
%similar examples.
%
%Let $p$ be an odd prime.  In this paper,
%for each $m \ge 2$,
%we give an example of finite $p$-group $G_m$ such that
%there exists
%\[0\not =\xi_{m}\in H_{nr}^{2m}(k(W)^{G_m};\bZ/p)\]
%which is not (multiplicatively) decomposed in the cohomology.
%(When $m=2$, $H_{nr}^i(k(W)^{G_2};\bZ/2)=0$ for $i<4$.)
%These groups are some central products  of an example given by
%Saltman and Bogomolov.   
% These groups are also central products of an elementary abelian
%$p$ group by the extraspecial $p$-group $p_+^{1+4m}$.
%
%
%To see the unramified cohomology, 
%we use the mod $p$ motivic cohomology $H^{*,*'}(BG;\bZ/p)$
%defined by Voevodsky
%of the classifying space $BG$.
%Let $\tau:H^{*,*'-1}(BG;\bZ/p)\to H^{*,*'}(BG;\bZ/p)$ be the map multiplying
%$0\not =\tau\in H^{0,1}(Spec(k);\bZ/p)\cong \bZ/p$.  We use the following
%injections
%\[H_{nr}^*(k(W)^G;\bZ/p)\subset H_{Zar}^0(BG;H_{\bZ/p}^*)\supset
%H^{*,*}(BG;\bZ/p)/(\tau)\]
%where $H_{\bZ/p}^*$ is the sheaf which associated the presheaf
%$U\mapsto H_{et}^*(U;\bZ/p)$.
%According Peyre (with some modifications), we define a subring
%$H_{nr}^*(G;\bZ/p)$ of $H^*(BG;\bZ/p)$ ($=H_{et}^*(BG;\bZ/p)$
%but it does not depend on fields $k$)
%such that its inflation image to $H^*(k(W)^G;\bZ/p)$ is in
%$H^*_{nr}(k(W)^G;\bZ/p)$. 
%
%We see $\xi_m\in H_{nr}^*(G;\bZ/p)$ by using the commutator calculus.
% We use the Milnor operation $Q_i$ defined
%by Voevodsky to see $\xi_m\not =0$ in $H^{*,*}(BG_m;\bZ/p)/(\tau)$.
%Indeed, $Q_{2m-1}...Q_0(\xi_m)\not=0$ in $H^*(BG_m;\bZ/p)$.






\section{motivic cohomology}

Let $X$ be a smooth (quasi projective) variety.
Let $H^{*,*'}(X;\bZ/p)$ be the $mod(p)$ motivic cohomology defined by Voevodsky and Suslin.



Recall that the ($ mod\ p$) $B(n,p)$ condition holds if 
\[H^{m,n}(X;\bZ/p)\cong H_{et}^m(X;\mu_p^{\otimes n})\qquad for\ all\ m \le n.\]
It is known that 
the $B(n,p)$ condition holds for $p=2$  or $n=2$ by Voevodsky([Vo1,2]), 
and
Merkurjev-Suslin respectively.
Quite recently M.Rost and V.Voevodsky ([Vo5],[Su-Jo],[Ro]) announced 
that $B(n,p)$ condition holds for
each $p$ and $n$. Hence the Bloch-Kato conjecture also holds.
Therefore in this paper, we $always$ $assume$ the $B(n,p)$-condition 
and so the Bloch-Kato conjecture for all $n,p$.


 Moreover we always assume that $k$ contains a primitive $p$-th $root$ $of$
 $unity$.
 For these cases, we see the isomorphism
$H_{et}^m(X;\mu _p^{\otimes n})\cong H_{et}^m(X;\bZ/p).$ 
Let $\tau$ be a generator of  $H^{0,1}(Spec(k);\bZ/p)\cong \bZ/p$.
Hence
\[ colim_{i}\tau^iH^{*,*'}(X;\bZ/p)\cong  H_{et}^*(X;\bZ/p).\]




Recall that $\bZ/p(n)$ ([Vo1,2,3]) is the 
complex of sheaves in Zarisky topology
such that
$H^{m,n}(X;\bZ/p)\cong H^m_{Zar}(X;\bZ/p(n)).$
Let $\alpha$ be the obvious map of sites from etale  topology 
to Zarisky  topology so that
\[H^{m}_{et}(X;\bZ/p)\cong H^{m}_{et}(X;\mu_p^{\otimes n})
\cong H^m_{Zar}(X,R\alpha_*\alpha^*\bZ/p(n)).\]
For $k\le n$, let $\tau_{\le k}R\alpha_*\alpha^*\bZ/p(n)$ be 
the canonical
truncation of $R\alpha_*\alpha^*\bZ/p(n)$ of level $k$. 
Then we have the short exact sequence of sheaves
 \[ \tau_{\le {n-1}}R\alpha_*\alpha^*\bZ/p(n)
 \to \tau_{\le n}R\alpha_*\alpha^*\bZ/p(n)\to 
  H_{\bZ/p}^n[-n]  \]
 where $H_{\bZ/p}^n$ is the Zarisky sheaf induced from the
  presheaf $H_{et}^n(V;\bZ/p)$ for open subset $V$ of $X$. 
The
Beilinson and Lichtenbaum conjecture ( hence $B(n,p)$-condition )
(see [Vo2,5]) implies
\[\bZ/p(k)\cong \tau_{\le k}R\alpha_*\alpha^*\bZ/p(n).\]
Hence we have ;
 \begin{lemma} ([Or-Vi-Vo], [Vo5]) There is the long exact sequence 
  \[  \to H^{m,n-1}(X;\bZ/p)\stackrel{\times \tau}{\to}
  H^{m,n}(X;\bZ/p)\qquad \qquad \qquad \]
  \[\qquad \qquad \to
H_{Zar}^{m-n}(X;H_{\bZ/p}^n)\to H^{m+1,n-1}(X;\bZ/p)\to .\]
\end{lemma}
 In particular, we have 
\begin{cor}The graded ring $grH^{m-n}_{Zar}(X;H_{\bZ/p}^n)$ is isomorphic to 
\[ H^{m,n}(X;\bZ/p)/(\tau)\oplus
Ker(\tau)|H^{m+1,n-1}(X;\bZ/p)\]
where $H^{m,n}(X;\bZ/p)/(\tau)=H^{m,n}(X;\bZ/p)/(\tau H^{m,n-1}(X;\bZ/p))$.
\end{cor}
Note that the above long exact sequence induces the $\tau$-Bockstein spectral sequence
\[E(\tau)_1=H^{m-n}_{Zar}(X;H_{\bZ/p}^n)\Longrightarrow colim_{i}\tau^iH^{*,*'}(X;\bZ/p)\cong  H_{et}^*(X;\bZ/p).
\] 







On the other hand, the filtration $coniveau$ is given by
\[N^cH^m_{et}(X;\bZ/p)=\cup_{Z} Ker\{H_{et}^m(X;\bZ/p) \to H^m_{et}(X-Z;\bZ/p)\}\]where $Z$ runs in the set of closed subschemes of $X$ of $codim=c$.
Grothendieck wrote down the $E_1$-term of the spectral sequence induced from the above coniveau filtration.
\[E(c)_1^{c,m-c} \cong \Pi _{x\in X^{(c)}}H_{et}^{m-c}(k(x);\bZ/p)
\Longrightarrow H_{et}^m(X;\bZ/p)\]
where $X^{(c)}$ is the set of primes of codimension $c$
and $k(x)$ is the residue field of $x$.
We can regard $i_{x*}H^{m-c}_{et}(k(x);\bZ/p)$ as a constant sheaf 
$H^{m-c}_{et}(k(x);\bZ/p)$ on
$\bar{\{x\}}$ and extend it by zero to $X$.  Then the differentials of the spectral sequence give us a complex on sheaves on $X$
\[(2.9)\ 0\to H_{\bZ/p}^q \to \Pi _{x\in X^{(0)}}i_{x*}H^q_{et}(k(x);\bZ/p)
\to \Pi_{x\in X^{(1)}}i_{x*}H^{q-1}_{et}(k(x);\bZ/p)\]
\[\qquad \qquad \to ...\to \Pi _{x\in X^{(q)}}i_{x*}H^0_{et}(k(x);\bZ/p)\to 0.\]
Bloch-Ogus [Bl-Og] proved that the above sequence of sheaves
is exact and the $E_2$-term is given by
\[E(c)_2^{c,m-c}\cong 
%\frac{Ker(\Pi _{x\in X^{(c)}}H^{c-m}_{et}(k(x);\bZ/p)
%\to \Pi_{x\in X^{(c+1)}}H^{m-1}_{et}(k(x);\bZ/p))}
%{Im(\Pi _{x\in X^{(c-1)}}H^{c-m+1}_{et}(k(x);\bZ/p)
%\to \Pi_{x\in X^{(c)}}H^{m}_{et}(k(x);\bZ/p))}\]
%\[\cong 
H_{Zar}^c(X,H_{\bZ/p}^{m-c}).\]

In particular, we have ;
\begin{cor}
\[ H^0_{Zar}(X;H_{\bZ/p}^m)\cong Ker\{H^m_{et}(k(X);\bZ/p)\to 
\Pi_{x\in X^{(1)}}H^{m-1}_{et}(k(x);\bZ/p)\}.\]
\end{cor}





By Deligne ( foot note (1) in Remark 6.4 in [Bl-Og]) and
Paranjape (Corollary 4.4 in [Pj]), it is proven that
there is an isomorphism of the coniveau spectral sequence
with the Leray spectral sequence for the map $\alpha$.
Hence we have ; 
   


\begin{thm} (Deligne, Parajape)
There is the isomorphism $E(c)_r^{c,m-c}\cong E(\tau)_{r-1}^{m,m-c}$
of spectral sequences.
Hence the filtrations are the same $N^cH_{et}^m(X;\bZ/p)= F_{\tau}^{m,m-c}$
where 
\[F_{\tau}^{m,m-c}=Im(\times \tau^c:H^{m.m-c}(X;\bZ/p)\to H^{m,m}(X;\bZ/p)).\]
\end{thm} 


















\section{cohomology of groups}

Let $G$ be a reductive algebraic group over $k$
acting on an affine variety $W$.
A point $x\in W$ is called stable if the orbit $Gx$ is closed and
the stabilizer group $Stab(x)$ is a finite group.
Let us write by $X^s$ the set of stable points in $X$.
Then $X^s$ is an open subset of $X$ and there is the commutative diagram
\[\begin{CD}
     W @>{\phi}>>  W//G  \\
     @AA{open}A    @AA{open}A \\
     W^s @>{\phi^{s}}>>  W^s/G,
     \end{CD} \]
     where $W//G=Spec(k[W]^G)$, the geometric quotient of $X$.
 We also note that the invariant field $k(W)^G$ is a quotient field
of $k[W]^G$ when  $X^s\not =\emptyset$.
%(6.2 in {Mukai]) 

Suppose $X^s\not =\emptyset$.  
Then $H^{*,*'}(W//G;\bZ/p)=H^{*,*'}(Spec(k[W]^G);\bZ/p)$ and 
$k(W^s/G)=k(W//G)=k(W)^G$. Hence we have the diagram
\[\begin{CD}
H^{*,*'}(W//G;\bZ/p)@>>> H^{*,*'}(Spec(k(W)^G);\bZ/p)\\
   @VVV  @V{=}VV    \\
H^{*,*'}(W^s/G;\bZ/p)@>{\psi}>> H^{*,*'}(Spec(k(W^s/G);\bZ/p)
\end{CD}\] 


Restrict $W$ as an affine space $W=\oplus k$ and let $\rho: G\to W=\oplus k$ a faithful representation.
Let $U_n=W-S$ be an open set of $W$ such that $G$ act freely $U$
where $codim_WS=n$.  (Of course $U$ is an open subset of $W^s$.) Then the 
classifying space of $G$ is defined as $colim_{n\to \infty} (U_n/G)$.
Then the $mod(p)$ motivic cohomology (for degree $*<2n$) of $BG$ is given by 
([Vo4],[To])
\[  H^{*,*'}(BG;\bZ/p)\cong lim_{n\to \infty}H^{*,*'}(U_n/G;\bZ/p).\]
In particular, by $BL(p,*)$ condition, we have
\[H^{*,*}(BG;\bZ/p)\cong H_{et}^*(BG;\bZ/p)= H^*(G;\bZ/p)\otimes H^*(k;\bZ/p)\]
where the last group is the cohomology group of the Galois group $G$
(when $G$ is finite).
Thus from the above diagram, we have the map
\[ \psi: H^{*,*'}(BG;\bZ/p)\to H^{*,*'}(Spec(k(W)^G);\bZ/p).\]

This  map $\psi^{*,*}=\psi_{et}^*$ is explains also as follows.
Let $\Gamma$ is the absolute Galois group $\Gamma=Gal(\bar {k(W)}/k(W)^G)$.  Then
the group $G=Gal(k(W)/k(W)^G)$ is a quotient group of the absolute Galois group $\Gamma$.
Then the map $\psi_{et}$ is the induced map from the quotient $q: \Gamma \to G$, i.e.,
\[ \psi_{et}^*=q^*:  H^*(G;\bZ/p)\to H^*(\Gamma;\bZ/p)=H^*(k(W)^G;\bZ/p).\]


\begin{lemma}
\[Im(\psi^*)\cong H^{*,*}(BG;\bZ/p)/(\tau).\]
\end{lemma}
\begin{proof}
For each field $F$, by the Bloch-Kato conjecture, $H^*(F;\bZ/p)$ is generated by
elements in $H_{et}^1(F;\bZ/p)\cong H^{1,1}(Spec(F);\bZ/p)$.
So
\[\psi^{*,*-1}: H^{*,*-1}(BG;\bZ/p)\to H^{*,*-1}(Spec(k(W)^G);\bZ/p)=0.\]
Hence the map $\psi^*$ is expressed as a composition
\[H^{*,*}(BG;\bZ/p)\to H^{*,*}(BG;\bZ/p)/(\tau)\to H^*_{et}(k(W)^G;\bZ/p).\]
The first map is of course surjective and we only need the injectivity of the second map.
Indeed, from Corollary 2.2 and 2.3, we see
\[H^{*,*}(BG;\bZ/p)/(\tau)\subset H^{0}_{Zar}(BG;H_{\bZ/p}^*)\subset H^*(k(W)^G;\bZ/p).\]
\end{proof}

Recall the coniveau filtration given in $\S 2$
\[N^cH^m_{et}(X;\bZ/p)=\cup_{Z} Ker\{H_{et}^m(X;\bZ/p) \to H^m_{et}(X-Z;\bZ/p)\}\]where $Z$ runs in the set of closed subschemes of $X$ of $codim=c$.  From Theorem 2.4, we see
\begin{cor}
$Im(q^*)\cong H^*_{et}(BG;\bZ/p)/(N^1H^*_{et}(BG;\bZ/p))$.
\end{cor}
 

According to Saltman (and Serre), we say an element $x\in H^*(G;\bZ/p)$ is geometrically negligible
if $\psi^*(x)=0$.  Let us write $Ng=Ng(G)=Ker(\psi^*)$.
From the above lemma, it is immediate
\[ Ng(G)=N^1H^*_{et}(BG;\bZ/p)\]
\[=Im(\times \tau|H^{*,*-1}(BG;\bZ/p)\to H^{*,*}(BG;\bZ/p))\]
and we have
\[ Im(\psi^*)=H^*(BG;\bZ/p)/(Ng)\cong H^*(BG;\bZ/p)/(N^1)\cong H^{*,*}(BG;\bZ/p)/(\tau) .\]
\begin{lemma}For each element $x$ in $H^*_{et}(BG;\bZ/p)$, the images of cohomology (Bockstein, reduced)
operations $\beta(x)$, $P^i(x)$ for $i>0$ are in $Ng(G)$ (hence $x^p\in Ng(G)$). 
The image of Gysin map $g_*(x)$ are also
in $Ng(G)$.
\end{lemma}
\begin{proof}
The cohomology operations act as ([Vo2,4])
\[\beta:H^{*,*}(X;\bZ/p)\to H^{*+1,*}(X;\bZ/p)\]
\[P^i:H^{*,*}(X;\bZ/p)\to H^{*+2i(p-1),*+i(p-1)}(X;\bZ/p).\]
For an element $x\in H^{m,n}(X;\bZ/p)$, define the difference degree
$d(x)=m-n$.  Then if $d(x)=0$, then $d(\beta(x))>0$ and $d(P^i(x))>0$.
Hence these elements are in $Im(\tau)$ as a subset of $H^{*,*}(X;\bZ/p)$.

For the embedding  $X\subset Y$ of codimension $c$, the Gysin map is defined on
\[ g_*: H^{*,*}(X;\bZ/p)\to H^{*+2c,*+c}(Y;\bZ/p).\]
By the same reason as the cases of cohomology operations, we get lemma.
\end{proof}

Here we give a sufficient condition for $x\not \in Ng(G)$.
Voevodsky define the Milnor operation $Q_i$ also in the mod $p$ motivic cohomology
\[Q_i:H^{*,*'}(-;\bZ/p)\to H^{*+2p^n-1,*'+p^n-1}(-;\bZ/p).\]
Define the weight $w(x)=2*'-*$ for element (or operation) $x\in H^{*,*'}(X;\bZ/p)$,
e.g., $w(\tau)=2$, $w(Q_i)=-1$ and $w(P^i)=0$. 
\begin{lemma}  Let 
  $x\in H^n_{et}(BG;\bZ/p)$ and $Q_{n-2}...Q_0(x)\not =0$.
Then $0\not =x \in H^{n}_{et}(BG;\bZ/p)/(Ng)$.
\end{lemma}
 \begin{proof}
 Identify $x$ as an element in $H^{n,n}(BG;\bZ/2)$.
Suppose that $x=\tau \bar x$.  So $w(\bar x)=n-2$ since $w(\tau)=2$.  Then $t_{\bC}(\bar x)=t_{\bC}(x)=x$
where $t_{\bC}: H^{*,*'}(X;\bZ/p)\to H^{*,*'}(X(\bC);\bZ/p)$ is the realization map.

The operation $Q_i$ descends the weight one. 
Let $\bar \psi=Q_{n-2}...Q_0(\bar x)$. Then $w(\bar \psi)=-1$
but $t_{\bC}(\bar \psi)=Q_{n-2}...Q_0(x)\not =0$.
This is a contradiction since $w(y)\ge 0$ for each nonzero element $y\in H^{*,*'}(Y;\bZ/p)$
and for smooth $Y$.
\end{proof}

We have the Kunneth formula for the etale cohomology of coefficient $\bZ/p$.  Since $Ng$ is an ideal, we have
the surjection 
\[H^*_{et}(BG_1;\bZ/p)/(Ng)\otimes_{H^*(k;\bZ/p)}H^*_{et}(BG_2;\bZ/p)/(Ng)\]
\[ \to H^*_{et}(B(G_1\times G_2);\bZ/p)/(Ng).\]
However it does not need isomorphic, because there is the possibility that
$x_1\otimes x_2\in Ng(G_1\times G_2)$ but $x_1\not \in Ng(G_1),\ x_2\not \in Ng(G_2)$.



\section{cohomology invariant}


Recall that $H^1(k;G)$ is the first non abelian Galois cohomology set of $G$, which represents the set
of $G$-torsors over $k$. It is very important to study $H^1(k;G)$, for example $H^1(k;O_n)$  is isomorphic to 
the set of isomorphism classes of non generate quadratic forms over $k$ of rank $n$. 
(For details, see the excellent book [Ga-Me-Se].)




The cohomology invariant is defined by 
\[ Inv^i(G,\bZ/p)=Func( H^1(F;G)\to H^i(F;\bZ/p))\]
where $Func$ means natural functions for each fields $F$ over $k$.
The cohomology invariant is studied by many authors.
The cohomology invariants $Inv^*(G;\bZ/p)$ are
computed (for example in [Ga-Me-Se])
for groups elementary abelian $2$-groups, $O_n$,$SO_n$, $G_2$,... 
It is also stated in [Ga-Me-Se] that for many $G$ (but not all)
$Inv^*(G;\bZ/p)$ are  detected by $Inv^*(H;\bZ/p)$ for elementary abelian $p$-subgroups $H$.

Let $x\in H^0(BG;H_{\bZ/p}^i)$. Given a $G$-torsor $E$ over $F$,
we can construct $x(E)\in H^i_{et}(F;\bZ/p)$.  Roughly speaking, we can identify $E$ as the 
pullback of some map
$f:Spec(F)\to BG$.  So we can take $x(E)=f^*(x)\in H^0(Spec(F);H^i_{\bZ/p})=H^i_{et}(F;\bZ/p)$. 
Indeed, Totaro proved [Ga-Me-Se] the following theorem in a letter to Serre.
\begin{thm}
$Inv^*(G;\bZ/p)\cong H^0(BG;H_{\bZ/p}^*).$
\end{thm}

Therefore we see
\begin{cor}
 $Im(\psi^*)\cong H^{*}(BG;\bZ/p)/(Ng)\subset Inv^*(G;\bZ/p).$
 \end{cor}



\section{abelian $p$-groups}


Let us write $H^{*,*'}=H^{*,*'}(Spec(k);\bZ/p)$ and $H^*=H^{*,*}=K_M^*(k)/p$
so that $H^{*,*'}\cong H^*[\tau].$
 First consider the case $G=\bZ/p^r$.  The $mod(p)$ motivic cohomology is computed
(as the case $\bZ/p$ in [Vo])
\[H^{*,*'}(B\bZ/p^r;\bZ/p)\cong H^{*,*'}[y(r)]\otimes \Lambda(x(r)) \quad |y(r)|=2,\ |x(r)|=1.\]
(When $p=2$ and $r=1$, we see by Voevodsky ([Vo2,4])
\[x(1)^2=\tau y(1)+\rho x(1)\quad with \ \rho=(-1)\in H^1=k^*/(k^*)^2.)\]


 
For the inclusion $i: \bZ/p^r \subset \bZ/p^s$ and quotient map $q: \bZ/p^s\to \bZ/p^r$, for $s\ge r$,
we have
\[i^*(y(s))=y(r),\ i^*(x(s))=0,\quad q^*(y(r))=0,\ q^*(x(r))=x(s).\]
Moreover we still know $x(r)\in H^{1,1}(BG;\bZ/p)$ and $y(r)\in H^{2,1}(BG;\bZ/p)$.
Thus we see  ($(Ker(\tau)|H^{*,*'}(B\bZ/p^r;\bZ/p)=0$).
\[Inv^*(\bZ/p^r;\bZ/p)\cong H^{*,*}(\bZ/p^r;\bZ/p)/(\tau) = H^*\{1,x(r)\}.\]

Next consider their product $G=\bZ/p^{r_1}\times ...\times \bZ/p^{r_s}$.  
The cohomology $H^{*,*'}(B\bZ/p^r;\bZ/p)$ has the Kunneth formula.
Hence the motivic cohomology
is given 
\[  H^{*,*'}(BG;\bZ/p)\cong H^{*,*'}[y(r_1),...,y(r_s)]\otimes \Lambda(x(r_1),..,x(r_s))\]
where $x(r_i)\in H^{1,1}(BG;\bZ/p)$ and $y(r_i)\in H^{2,1}(BG;\bZ/p)$.

Recall that $H^*(BG;\bZ/p)/(Ng)\cong H^{*,*}(BG;\bZ/p)/(\tau)$ (Lemma 3.1).
Then we get
\begin{lemma} Let $G$ be an abelian $p$-group, i.e.,  $G=\oplus_i \bZ/(p^{r_i})$.  Then
\[ Inv^*(G;\bZ/p)\cong H^{*}(G;\bZ/p)/(Ng)\cong H^{*}\otimes\Lambda(x(r_1),...,x(r_s))\]
\[when\ p=2\ r_i=1,\quad x(r_i)^2=\rho x(r_i).\]
\end{lemma}
The elementary $2$-groups cases are stated in 
Theorem 16.4 in [Ga-Me-Se].

The $Q_i$-operation acts on $H^{*,*'}(B\bZ/p;\bZ/p)$ by $Q_i(x)=y^{p^i}$ (while $Q_i(x(j))=0$ for all $j>1$).
We consider $Q_i$ action on 
\[H^{*,*'}(B(\bZ/p)^s;\bZ/p)\cong H^{*,*'}[y_1,...,y_s]\otimes \Lambda(x_1,...,x_s).\]
Each $Q_i$ is a derivation, and hence
\[Q_0...Q_{s-1}( x_{1}...x_{s})=
\sum sgn(j_1,...,j_s) y_{1}^{p^{j_1}}y_2^{p^{j_2}}...y_{s}^{p^{j_s}}\not =0\]
where $(j_1,...,j_s)$ are permutations of $(0,...,s-1)$.
Thus we see that this case satisfies the sufficient  condition of Lemma 3.3
while the other cases does not), in fact $x_1...x_s\not \in Ng(G)$.

Let us say that an element $x\in H^{*}(BG;\bZ/p)/(Ng)$ is $detected$
 by an elemntary abelian $p$-subgroup $A$ if
 $Res(x)\not =0$ for
\[ Res: H^*(BG;\bZ/p)/(Ng) \to  H^*(BA;\bZ/p)/(Ng).\]
 The following lemma is immediate from the above arguments.
 \begin{lemma}
 If $x\in H^n(BG;\bZ/p)/(Ng)$ is detected by elementary abelian $p$-subgroups,
 then $Q_{n-1}...Q_0(x)\not =0$ in $H^*(BG;\bZ/p)$.
 \end{lemma}









\section{Cases $G=O_n$ and $SO_n$}

Hereafter, in this paper (except for  $\S 11$), we assume that 
$k=\bC$ otherwise stated.



lt is well known that
\[H^{*}(BO_n;\bZ/2)\cong H^{*}((B\bZ/2)^{\times n};\bZ/p)^{S_n}\cong \bZ/2[w_1,...,w_n]\]
where $S_n$ is the $n$-th symmetric group, and $w_i$ is the Stiefel-Whitney class representing the $i$-the elementary symmetric function.
we easily see
\[Q_{i-1}...Q_{0}(w_i)=y_1^{p^{i-1}}y_2^{p^{p-2}}...y_i+...\not =0
  \in \bZ/2[y_1,...,y_n]\subset H^*(B(\bZ/2)^n;\bZ/2)\]
 and hence $w_i\not \in Ng(G)$.
Recall the Wu formula
\[Sq^iw_k=\sum_j^i \left( \begin{array}{c}
{k-j-1}\\  {i-j}\end{array}\right) w_{k+i-j}w_j\quad (0\le i\le k).\]
Many cases of product of $w_iw_j$ are in $Ng(G)$, e.g.,$w_i^2\in Ng(G)$. 
More precisely, the motivic cohomology of $BO_n$ is computed for $k=\bC$ (Theorem 8.1 in [Ya3]) 
\[H^{*,*'}(BO_n;\bZ/2)\cong H^{*,*'}((B\bZ/2)^{\times n};\bZ/p)^{S_n}.\]

Given $x\in H^*(BG;\bZ/p)$,  let us define the weight $w(x)$ as the smallest weight $w(x')$ 
such that $t_{\bC}(x')=x$ with $x'\in H^{*,*'}(BG;\bZ/p)$. 
Indeed,  the weight of the symmetric polynomial 
\[t=\sum x_1^{2i_1+1}...x_{k}^{2k+1}x_{k+1}^{2j_1}...x_{k+q}^{2j_q}\quad in\  H^*((B\bZ/2)^{\times n};\bZ/2)\]
(with $0\le i_1\le ...\le i_k$, $0\le j_1\le...\le j_q$)
is given by $w(t)=k$.
Hence  if $t\not \in Ng$, then
$w(t)=deg(t)$ and this implies $t=x_1...x_i=w_i$.  
\begin{thm} 
\[Inv^*(O_n;\bZ/2)\cong H^*(BO_n;\bZ/2)/(Ng)\cong \bZ/2\{1,w_1,...,w_n\}.\]
\end{thm}
In fact, $Inv(O_n;\bZ/2)$ is well known (Theorem 17.3 in [Ga-Me-Se]) for general $k$ ;
\[Inv^*(O_n;\bZ/2)\cong H^*\{1,w_1,...,w_n\}.\]

We consider the multiplicative structure of $Inv^*(O_n;\bZ/2)$.
From the Wu formula, we see
   \[ Sq^1(w_{2i})=w_{2i+1}+w_{2i}w_1\in Ng(O_n).\]
   Hence $w_{2i+1}=w_{2i}w_1$ in $Inv^*(O_n;\bZ/2)$.
   By Rost and Kahn [Ka], the divided power operation can be defined in $K_*^M(F)/p$
compatible with  fields $F$ over $k$ 
(and hence $Inv^*(G;\bZ/p)$) if $\surd{-1}\in k$. 
Vial showed  [Via] that the divided power operations are only compatible maps
(natural maps) with field extensions over $k$.  
Moreover Becher [Be] showed that $\gamma_n(w_2)=w_{2n}$. 
(See also Milnor  p133 in [Mi].)
\begin{thm} (Becher [Be], [Via]) Let $\surd{-1}\in k$.  
Then $Inv^*(O_n;\bZ/2)$ is generated by $w_1$ and $w_2$ as an $H^*$-ring with
divided powers by
\[ \gamma_i(w_2)=w_{2i},\quad w_{2i+1}=w_{2i}w_1.\]
\end{thm}

Next consider the case $G=SO_n$.  It is well known that
\[H^*(BSO_n;\bZ/2)\cong H^*(BO_n;\bZ/2)/(w_1)\cong \bZ/2[w_2,...,w_n].\]
Let $n=2m+1=odd$.
For this case, there is the isomorphism
\[O_{2m+1}\cong SO_{2m+1}\times \bZ/2.\]
Let $p:O_n\to SO_n$ is the projection and $i:SO_n\to O_n$ the inclusion.  We consider the induced map
$p^*$ and $i^*$ on the mod $2$ motivic cohomology of their classifying spaces.  
Since $p^*(w_1)=0$, we see $w_{2i+1}\in Ng(G)$ from the above theorem
(in fact, $Sq^1(w_{2i})=w_{2i+1}$ in $H^*(BSO_n;\bZ/2)$).  Moreover $i^*w_{2i}=w_{2i}\ mod (Ng(G))$.
Thus we have
\begin{thm}
For $G=SO_{2m+1}$, we have
\[Inv^*(G;\bZ/2)\cong H^{*}(BG;\bZ/2)/(Ng)\cong \bZ/2\{1,w_2,...,w_{2m}\}.\]
\end{thm} 
Moreover $Inv^*(SO_{2m+1};\bZ/2)$ is still computed (Theorem 19.1 in [Ga-Me-Se]) for general $k$ ;
\[Inv^*(SO_{2m+1};\bZ/2)\cong H^*\{1,w_2,...,w_{2m}\}.\]

Now consider the case $n=2m=even$.  This case the mod $2$ motivic cohomology is not
computed even $k=\bC$ for $n>4$. However we compute $H^{*,*}(BSO_n;\bZ/2)/(\tau)$
easily.  Consider the inclusion
\[ SO_{2m-1}\stackrel{i_1}{\to}SO_{2m}\stackrel{i_2}{\to}SO_{2m+1}.\]
Since the restriction map for $i<m$
\[i_1^*(w_{2i})\not =0 \in H^{*,*}(BSO_{2m-1};\bZ/2),\]
we see $w_{2i}\not \in Ng(SO_{2m})$.  
Moreover we know that $w(w_{n})=n-2$ in Lemma 9.2 in [Ya3].
On the other hand, the monomial $w_I$ of not $w_{2i}$
are all in $Ng(SO_{2m+1})$ and so  $i_2^*(w_I)\in Ng(SO_{2m})$.

Thus we have
\begin{thm}
$H^{*}(BSO_{2m};\bZ/2)/(Ng)\cong \bZ/2\{1,w_2,...,w_{2m-2}\}.$
\end{thm}

Next we study $Inv^*(SO_{2m};\bZ/2)$.
There is an element (Lemma 9.3 in [Ya3]) in the motivic cohomology
\[ u_{m-1}\in H^{n,n-2}(BSO_n;\bZ/2)\quad with \ \tau u_{m-1}=0.\] 
So there is the nonzero element
\[ u\in Ker(\tau|H^{n,n-2}(BG;\bZ/2))\subset H^0(BG;H_{\bZ/2}^{n-1}).\]

On the other hand in [Ga-Me-Su], it is proved (Theorem 20.6) for general $k$
\[ Inv^*(SO_{2m};\bZ/2)\cong H^*\{1,w_2,...,w_{n-2}\}\oplus (Im(I_{\delta})).\]
Here when $k=\bC$, $Im(I_{\delta})\cong \bZ/2\{u\}$ with 
$deg(u)=n-1$ from Proposition 20.1 in [Ga-Me-Se].  Thus we see
\begin{thm}
Let $G=SO_{2m}$and $m\ge 2$.  Then for $deg(u)=2m-1$,
\[Inv^*(G;\bZ/2)\cong H^{*}(BG;\bZ/2)/(Ng)\oplus \bZ/2\{u\} .\] 
\end{thm}

From 22.10 in [Ga-Me-Se], it is known that
 \[Res: Inv^*(G;\bZ/p)\to Inv^*(H;\bZ/p)\]
 is injective for $p=2$, $G=SO_{n}$ and $H=(\bZ/2)^{n-1}$.  Hence
 for
 \[Res: H^0(BSO_{n};H_{\bZ/2}^*)\to H^0(B(\bZ/2)^{n-1};H_{\bZ/2}^*)\]
we have $Res(u)=x_1...x_{n-1}$.  Of course $u\not \in H^{*,*}(BSO_n;\bZ/2)/(\tau)$
but $x_1...x_{n-1}\in H^{*,*}(B(\bZ/2)^{n-1};\bZ/2)/(\tau)$.

Recall that in Corollary 2.2 $grH^0(BG;H^*_{\bZ/p})$ is defined by the filtration
$H^{*,*}(BG;\bZ/p)/(\tau)\subset H^0(BG;H^*_{\bZ/p})$.
So note that 
\[Res: grH^0(BSO_{n};H_{\bZ/2}^*)\to grH^0(B(\bZ/2)^{n-1};H_{\bZ/2}^*)\]
is not injective (in Corollary 2.2).



\section{$Spin_n$ and exceptional groups }

The $mod(2)$ cohomology of $BSpin_n$ is computed by Quillen
\[H^*(BSpin_n;\bZ/2)\cong  \bZ/2[w_{2^h}(\Delta)]\otimes \bZ/2[w_2,...,w_n]/(Q_iw_2|0\le i\le h)\]
where $w_i(\Delta)$ (resp. $w_i$)  is the Stiefel-Whitney class of a spin representation $\Delta$
(resp. usual representation $Spin_n\to SO_n$),  and $2^h$ is the the Radon-Hurwitz number (See [Qu] p.210).
By the result of Becher (Theorem 6.2), we have
\begin{thm} $H^{*}(BSpin_n;\bZ/2)/(Ng)=\bZ/2$  for $n>4$.
\end{thm}
\begin{proof}
Let us write representations $j:Spin_n\to SO_n$ and $\Delta:Spin_n\to SO_N$.
We consider the induced map in Galois cohomolgy
\[ j^* :H^{*}_{et}(k(W)^{SO_n};\bZ/2)\to H_{et}^*(k(W)^{Spin_n};\bZ/2).\]
By the Quillen's result, we see $j^*(w_2)=0$.  By Rost and Kahn [Ka], the divided powers 
naturally act on $K_*^M(F)/p$ for field $F$ over $k$.  Hence from Becher theorem
(Theorem 6.2),we get
\[ j^*(w_{2i})=j^*(\gamma_i(w_2))=\gamma_i(j^*(w_2))=0.\]
Similarly $w_2(\Delta)=0$ implies $w_{2^{h}}(\Delta)=0$  if $n>4$.
\end{proof}
\begin{cor}  For $*>0$, there is the isomorphism
\[Inv^*(Spin_n;\bZ/2)\cong Ker(\tau)|H^{*+1,*-1}(BSpin_n;\bZ/2).\]
\end{cor}




The $mod(2)$ motivic cohomology of $BSpin_7$ is computed in (Theorem 9.6 in [Ya3]).
We can easily see the above theorem also from the concrete computation.
Moreover there are $\tau$-torsion elements
\[ y_2\in H^{4,2}(Spin_7;\bZ/2),\quad y_2'\in H^{5,3}(Bspin_7;\bZ/2).\]
Therefore we can take $u\in H^0(BSpin_7,H_{\bZ/2}^3), v\in H^0(BSpin_7,H_{\bZ/2}^4)$.
\begin{thm}  \[Inv^*(BSpin_7;\bZ/2)\cong \bZ/2\{1,u,v\}\quad |u|=3,\ |v|=4.\]
\end{thm}

We consider exceptional Lie group types  $G_2,F_4$ and split $E_6$.
In [Ga-Me-Su], it is proved (Teorem 18.1,  Theorem 22.5)
\[Inv^*(G_2;\bZ/2)\cong Inv^*(E_6;\bZ/2)\cong H^*\{1,u\},\]
\[Inv^*(F_4;\bZ/2)\cong H^*\{1,u,f_5\}.\]
(Unfortunately, we can not reexplain $f_5$ by using $H^{*,*'}(BF_4;\bZ/2)$,
which is not computed yet.)

Moreover restriction image for elementary abelin $2$-subgroup of rank $3$
(resp. rank $5$) is injective for $G_2$, $E_6$ (resp. for $F_4$),
see 22.10 in [Ga-Me-Se].
\begin{thm} Let $G=G_2,F_4$. 
Then $H^*(BG;\bZ/2)/(Ng)=\bZ/2.$
\end{thm}
\begin{proof} It is known that the inclusion $i: G_2\to Spin_7$ induces the epimorphism
$i^*: H^*(BSpin_7;\bZ/2)\to H^*(BG_2;\bZ/2)$. Hence the result for $G=G_2$ follows from
$H^*(BSpin_7;\bZ/2)/(Ng)=\bZ/2$.

It is also known that the inclusion $i':Spin_9\to F_4$ induces the injection
${i'}^*:H^*(BF_4;\bZ/2)\to H^*(BSpin_9;\bZ/2)$.  The groups $Spin_9$ and $F_4$ has
the same maximal abelin $2$-group  $H$ (of rank 5).  We consider the restriction to this $H$.
The fact $Res(H^*(BSpin_9;\bZ/2)/(Ng))=\bZ/2$ implies the results for $F_4$, because
the restriction is injective from [Ga-Me-Se].
\end{proof}

Here we give an example.  The $mod(2)$ cohomology is well known
\[H^*(BG_2;\bZ/2)\cong \bZ/2[w_4,w_6,w_7].\]
The motivic cohomology is given in  Theorem 7.5 in [Ya3].
In particular, $c_4,c_6,c_7$ are  the Chern classes (in $H^{2*,*}(BG_2;\bZ/2)$) so that
$\tau^{i}c_i=w_i^2$.
Let us write simply by $Q(n)$ the exterior algebra $\Lambda(Q_0,...,Q_n)$.
\begin{thm} The motivic cohomology $H^{*,*'}(BG_2;\bZ/2)$ is isomorphic to 
\[   \bZ/2[c_6,c_4]\otimes (\bZ/2\{y_2\} \oplus \bZ/2[\tau]\otimes
(\bZ/2\{1\} 
\oplus (\bZ/2[c_7]Q(2)-\bZ/2\{1\})\{a\})\]
where $deg(y)=(4,2)$, and $a$ is a virtual element with $deg(a)=(3,3)$ so that $c_7a=w_6w_7w_4$,
$Q_0a=w_4$, $Q_1a=w_6$ and $Q_2a=w_6w_4$.
\end{thm}
Let us write $Q_{i_1}...Q_{i_s}(a)$ by $Q_{i_1...i_s}$ and $w_{i_1}...w_{i_s}$ by
$w_{i_1...i_s}$.  Then
$(\bZ/2[c_7]Q(2)-\bZ/2\{1\})\{a\}$ is written as
\[ \bZ/2[c_7]\{c_7a=w_{467}, Q_0=w_4,Q_1=w_6, Q_{01}=w_7,\]
\[ Q_2=w_{46},Q_{02}=w_{47},
Q_{12}=w_{67},Q_{012}=c_7\}.\] 
Note that all $w_{i_1...i_s}$ (which are generators of the $\bZ/2[c_4,c_6,c_7]$-module
$H^*(G_2;\bZ/2)$) appeared indeed in the above. Moreover it is immediate that 
all elements except for $1$ and $a$ are in $Ng(G_2)$, and $y\in Ker(\tau)|H^{4,2}(BG_2;\bZ/2)$.



  

Next we consider odd prime case.  In 22.10 in [Ga-Me-Se]
it is stated that the restriction image of $Inv^*(G;\bZ/p)$ to some (maximal) 
elementary abelian $p$-group $H$ is injective for $(G,p)=(F_4,3),(E_6,3),(E_7,3)$ or $(E_8,5)$.

In these cases, each exceptional Lie group has two conjugacy classes of maximal
elementary abelian $p$-groups. One is the subgroup of a maximal torus
and the other is a nontoral $A$.  Let us write $i_A:A\to G$ and $i_T:T\to B$
be the inclusions. Tezuka and Kameko proves ([Ka], [Ka-Ya]) that  the following map is injective
\[i_A^*\times i_T^*:H^*(BG;\bZ/p)\to H^*(BA;\bZ/p)\times H^*(BT;\bZ/p),\]
namely, $H^*(BG;\bZ/p)$ is detected by $A$ and $T$.
Since $H^*(BT;\bZ/p)$ is non-nilpotent, the above group $H$  must be $A$. 
\begin{thm}
Let $(G,p)=(F_4,3),(E_6,3),(E_7,3)$ or $(E_8,5)$. Then
$H^*(BG;\bZ/p)/(Ng)=\bZ/p.$
\end{thm}
\begin{proof}
The restriciton image to $A$ is studied in [Ka-Ya].  
Images are generated as a ring by Chern classes and
\[Q_I(x_1..x_s) \in  H^*(BA;\bZ/p)\cong \bZ/p[y_1,...,y_s]\otimes \Lambda(x_1,...,x_s)\]
 where $Q_I=Q_{i_1}...Q_{is}$  for some $I\not =\emptyset$.  (Note $x_1...x_s\not \in Im(i_A^*)$.)
 Hence $i_A^*(H^+(BG;\bZ/p))\in Ng(A)$.
\end{proof}
 Here we give an example.   The mod $3$ cohomology of $BF_4$ is completely determined by Toda.
\begin{thm}([Toda])  The cohomology 
$H^*(BF_4;\bZ/3)$ is isomorphic to 
\[\bZ/3[x_{36},x_{48}]\otimes (\bZ/3[x_4,x_8]\otimes\{1,x_{20},x_{20}^2\}+ \bZ/3[x_{26}]\otimes \Lambda(x_9)\otimes \{1,x_{20},x_{21},x_{25}\})\]
where  the above two terms have the intersection $\{1,x_{20}\}$. 
\end{thm}
Indeed, we see that 
 $x_{26}|A=Q_0Q_1Q_2(u_3),\  x_{36}|A=c_{3,1},\  x_{48}|A=c_{3,2},
 \ x_4|A=Q_0(u_3),\ 
x_8|A=Q_1(u_3), \ x_{20}|A=Q_2(u_3)$, $x_9|A=Q_0Q_1(u_3),$
$x_{21}|A=Q_0Q_2(u_3),$  $x_{25}|A=Q_1Q_2(u_3)$.
Here $u_3=x_1x_2x_3$ and $x_{36}$ and $x_{48}$ are  represented by Chern classes.




\section{  $GL_n(\bF_{\ell})$ and $PGL_p$}


Of course, there is the isomorphism  
\[H^{*,*'}(BGL_n;\bZ/p)\cong H^{*,*'}[c_1,...,c_n]\]
where $c_i$ is the Chern class.  Hence $Inv^*(GL_n;\bZ/p)\cong \bZ/p$
(for $k=\bC$).
Let $G$ be a finite group such that
\[ (8.1) \quad H^*(BG;\bZ/p)\cong \bZ/p[c_{i_1},...,c_{i_n}]\otimes \Lambda(e_{i_1},...,e_{i_n})\]
where $\beta_{j_s}(e_{i_s})=c_{i_s}$ where $\beta_{j_s}$ is the higher Bockstein operation,
and $c_{i_s}$ is a Chern class of some representation of $G$,
e.g.,  $G=GL_n(\bF_{\ell})$ where $\ell$ is prime to $p$.
\begin{thm}
Let $G$ be a finite group given as (8.1) so that $i_1\ge 2$.  Then
$H^*(BG;\bZ/p)/(Ng)\cong \bZ/p$.
\end{thm}
\begin{proof}
First note that $H^*(BG;\bZ/p)/(Ng)$ is a quotient $\Lambda(e_{i_1},...,e_{i_n})$.
Since the motivic cohomology has the transfer map,  we know that each element $x$ in
$H^{*,*'}(BG;\bZ)$ has the exponent dividing $|G|$.
Hence there is $e_s'\in H^{*,*-1}(BG;\bZ/p)$ such that 
\[\beta_{s'}(e_s')=\tau^{i_s-2}c_{i_s}.\]
From Lemma 2.1, we know that $\tau:H^{*,*-1}(X;\bZ/p)\to H^{*,*}(X;\bZ/p)$ is injective.
Hence $\tau e_s'\not =0$.  This means $\beta_{s''}(\tau e_s')=\tau^{i_s-1}c_{i_s}$
for $s''\le s'$.  So $s''=i_s$ and we can take $\tau e_s'=e_{i_s}$ 
$mod (Ideal(e_{i_1},...,e_{i_{s-1}})).$
By induction on $s$, we can prove all $e_{i_s}\in Ng$.
\end{proof}




 
 Let $p$ be an odd prime and denote by $PGL_p$ the projective
 group which is the quotient of the general linear group
 $GL_p$ by the center $\bG_m$.   Its ordinary $mod(p)$ cohomology  and the 
Chow ring are known by Vistoli \cite{Vi}
and Kameko-Yagita [Ka-Ya].
 
 To state the cohomology $H^*(BPGL_{p};\bZ/p)$, we recall the Dickson 
algebra. Let $A\cong (\bZ/p)^n$ be  an elementary abelian $p$-group of rank $n$, and
$H^*(BA)\cong \bZ/p[y_1,...,y_n]\otimes \Lambda(x_1,...,x_n)$.
 The Dickson algebra is 
\[ D_n=\bZ/p[y_1,...,y_n]^{GL_n({\bF}_p)}\cong 
{\bZ/p}[c_{n,0},...,c_{n,n-1}]\]
with $|c_{n,i}|=2(p^n-p^i)$.  The invariant ring under $SL_n({\bF}_p)$
is also given
\[SD_n=\bZ/p[y_1,...,y_n]^{SL_n(\bF_p)}\cong 
D_n\{1,e_n,...,e_n^{p-2}\}\quad with\ e_n^{p-1}=c_{n,0}.\]
We also recall the Mui's result by using $Q_i$ 
according to Kameko and Mimura  \cite{Ka-Mi}
\[gr H^*(BA)^{SL_n(\bF_p)}\cong 
SD_n/(e_n)\oplus SD_n\otimes Q(n-1)\{u_n\}\]
where $u_n=x_1...x_n$ and $e_n=Q_0...Q_{n-1}u_n$.
(Here note
\[SD_n/(e_n)\cong D_n/(c_{n,0})\cong \bZ/p[c_{n,1},...,c_{n,n-1}].)\]
\begin{thm}(\cite{Vi},\cite{Ka-Ya}) There is the isomorphism
\[H^*(BPGL_p;\bZ/p)\cong M\oplus N\]
where 
$M\cong \bZ/p[x_{4}, x_{6}, \cdots , x_{2p}]$
as modules (but not rings) and
\[N\cong SD_2\otimes Q(1)\{u_2\}
\cong \bZ/p[e_2,c_{2,1}] \otimes Q(1)\{u_2\}.\]
\end{thm}
\begin{thm} (\cite{Vi},\cite{Ka-Ya})
There is the additive isomorphism
\[CH^{*}(BPGL_p)/p\cong M \oplus SD_2\{Q_0Q_1u_2\}.\]
\end{thm}
It is  also proved that $Ker(\tau)|H^{*,*'}(BPGL_p;\bZ/p)=0$ in
Theorem 10.4 in [Ya3].
\begin{thm}   
\[Inv^*(PGL_p;\bZ/p)\cong H^*(G;\bZ/p)/(Ng)\cong \bZ/2\{1,u\}    \quad |u|=2.\]
\end{thm}
This fact is also shown in [Ga-Me-Se].



\section{Symmetric group $S_n$}


Let $S_n$ be the Symmetric group generated by permutations of $n$-letters.
The permutations induce the natural representation $S_n\to O_n$.
Let us write by $w_i$ its Stiefel-Whitney class.  Then it is proved
for example in [Ga-Me-Se] that for general $k$
\begin{thm}
$Inv^*(S_n;\bZ/2)\cong  H^*\{1,w_1,...,w_{[n/2]}\}.$
\end{thm}
Let $A$ be a subgroup of $S_n$ generated by the transpositions
$(2i-1,2j)$ for $1\le j\le [n/2]$ so that
$A\cong \oplus ^{[n/2]}\bZ/2$.  Then $Inv^*(S_n;\bZ/2)$ is 
detected by the group $A$.

In this section, by using the cohomology $H^*(BS_n;\bZ/2)$,
we will reexplain above facts (for $H^*(BS_n;\bZ/2)$) but when  $k=\bC$ and $n=2^m$.
(We assume $k=\bC$.)

V.Voevodsky showed (for the definition of the reduced power operation) 
\[H^{*,*'}(BS_p;\bZ/p)\cong H^{*,*'}(B\bZ/p;\bZ/p)^{S_p}\cong \bZ/p[Y]\otimes \Lambda(X)\]
where $Y=y^{p-1}$ and $X=y^{p-2}x$ in $H^{*,*'}(B\bZ/p;\bZ/p)\cong \bZ/p[\tau, y]\otimes \Lambda(x).$
Hence $H^*(BS_p;\bZ/p)/(Ng)=\bZ/p$ for $p\not =2$  but $H^*(BS_2;\bZ/2)/(Ng)\cong \Lambda(x)$.

Now we restrict $p=2$ and let $n=2^m$.
Consider the natural embedding
\[ g_m: V_m=\oplus ^m\bZ/2 \to \bZ/2\wr...\wr\bZ/2 \to S_{2^m}\]
where $-\wr-$ is the wreath product.
Then it is known (Theorem 3.23 in [Ma-Mi]) that $H^*(BS_{2^m};\bZ/2)$ is detected
by $S_{2^{m-1}}\times S_{2^{m-1}}$ and $V_m$.

Let $f_m: V_m\stackrel{g_m}{\to}S_n\to O_n$.  The the total Whitney class is written
as
\[ w( f_m)=1+w(m-1)+w(m-2)+...+w(0)\]
where $w(i)=w_{2^m-2^i}(f_m)$.  Moreover $Sq^{2^i}(w(i+1))=w(i)$.
In fact the image of $g_m^*$ is just the Dickson algebra
\[ Im(g_m^*)=\bZ/2[x_1,...,x_m]^{GL_2(\bF_2)}=\bZ/2[w(m-1),....,w(0)].\]
( Here $w(i)^2$ corresponds to $c_{m,i}$ for odd prime cases stated in the preceding section.)

By induction on $m$, we assume
\[H^*(BS_{2^{m-1}};\bZ/2)/(Ng)\cong \bZ/2\{1,w_1,...,w_{2^{m-2}}\}.\]
Then considering the restriction
\[ H^*(BS_n;\bZ/2)/(Ng)\to H^*(BS_{n-1};\bZ/2)/(Ng)\otimes H^*(BS_{n-1};\bZ/2)/(Ng),\]
we see
\[ H^*(BS_n;\bZ/2)/(Ng)\supset \bZ/2\{1,w_1,...,w_{2^{m-2}},...,w_{2^{m-1}}\}.\]
Note by construction of maps $g_m,f_m$, we see $g_m^*w_{2^{m-1}}=w(m-1)$.
Hence
\[ w(j)'=Sq^{2^j}...Sq^{2^{m-2}}w_{2^{m-1}}\in Ng(S_n)\]
with $g_m^*(w(j)')=w(j).$  Thus we have
\begin{prop}  
  $ H^*(BS_n;\bZ/2)/(Ng)\cong \bZ/2\{1,w_1,...,w_{2^{m-1}}\}.$
\end{prop}

In [Ga-Me-Se], it is also shown $Inv^*(S_n;\bZ/p)=\bZ/p$  for odd prime $p$.
Let $n=p^m$.  The symmetric group $S_n$ has a subgroup isomorphic to
  \[S(m)=\bZ/p\wr S_{p^{m-1}}\cong (S_{p^{m-1}})^p\rhd \bZ/p\]
 of the index prime to $p$.  Hence $H^*(BS_n;\bZ/p)\subset H^*(BS(m);\bZ/p)$.
 
We consider the Hochshild-Serre spectral sequence
\[ E_2^{*,*'}=H^*(\bZ/p;H^*((BS_{p^{m-1}})^p;\bZ/p))\Longrightarrow
H^*(BS(m);\bZ/p).\]

Let us write by $\sigma$ the generator of the cyclic group
$\bZ/p$.  Let $T=(1-\sigma)$ and $N=(1+\sigma+...+\sigma^{p-1})$.
For a $\bZ/p$-module $M$, the homology is written
\[H^*(\bZ/p; M)=\begin{cases}
Ker(T)=M^{\bZ/p}\qquad *=0\\
   Ker(T)/Im(N)\qquad *=even>0\\
   Ker(N)/Im(T)\qquad *=odd.
   \end{cases}\]
   Let $\{x_i\}$ be a $\bZ/p$ basis of $H^*(BS_{p^{m-1}};\bZ/p)$.
Then the basis of 
\[S=H^*((BS_{p^{m-1}})^p;\bZ/p)\cong \bZ/p\{x_{i_1}\otimes x_{i_2}\otimes....x_{i_p}\}.\]
  decomposes as  $I\cup F$ with 
   $I=\{x_i\otimes ...\otimes x_i\}$
and 
\[   F=\{x_{i_1}\otimes ...\otimes x_{i_s}| i_k\not =i_\ell\ for \ \ some\ k\not =\ell \}.\]
   The generator $\sigma$ acts on $F$ freely but invariants on $I$.
    Then cohomology is computed
   \[H^*(\bZ/p;\bZ/p\{I\})\cong \bZ/p[y]\otimes \Lambda(x)
   \otimes \bZ/p\{I\}
   \quad |x|=1,\ |y|=2\]
   \[H^*(\bZ/p;\bZ/p\{F\})\cong \bZ/p\{F\}^{\bZ/p}.\]
     Since $S^{\bZ/p}=\bZ/p\{F\}^{\bZ/p}\oplus \bZ/p\{I\}$,
   we have the isomorphism
   \[E_2^{*,*'}\cong S^{\bZ/p}\oplus \bZ/p\{I\}\otimes \bZ/p[y]\{x,y\}.\]
      It is well known that $S^{\bZ/p}\subset H^*(BG;\bZ/p)$
   by Nakaoka (also Totaro \cite{To})).  Hence this spectral sequence
   collapses from the $E_2$-term.  Thus we have
   the  well known result ;
   \begin{thm}(Nakaoka)
   \[H^*(BS(m);\bZ/p)\cong 
   E_2^{*,*'}\cong S^{\bZ/p}\oplus  H^*(BS_{p^{m-1}};\bZ/p)^{[p]}\otimes \bZ/p[y]\{x,y\}\]
   where $A^{[p]}$ is the graded algebra whose degree is given by $p$-times of degree of $A$.
   \end{thm}
   By induction on $m$, we may assume 
$H^*(BS_{p^{m-1}};\bZ/p)/(Ng)=\bZ/p$.  Hence $\bZ/p\{I\}/(Ng)=\bZ/p$.
Therefore there is the surjection 
\[ \bZ/p\{1,x\}\to H^*(BS;\bZ/p)/(Ng).\]
But $H^2(BS_n;\bZ/p)=0$. Hence $H^+(BS_n;\bZ/p)/(Ng)=0$.
\begin{prop}
       $H^*(BS_{p^m};\bZ/p)/(Ng)=\bZ/p$.
       \end{prop}
       
       We note here about  the restriction image for
$g_m: V_m=\bZ/p\to S_{n}$. The restriction image is contained in the Dickson algebra
as stated in the preceding section 
\[gr H^*(BA)^{GL_n(\bF_p)}\cong 
D_n/(c_{n,0})\oplus D_n\otimes Q(n-1)\{e_n^{p-2}u_n\}\]
where $u_n=x_1...x_n$ and $e_n=Q_0...Q_{n-1}u_n$.
(Here note
$D_n/(c_{n,0})= \bZ/p[c_{n,1},...,c_{n,n-1}].$)
We know $D_n$ is in the image of $g_m^*$.  However it is known that
 $e_n^{p-2}u_n\not \in g_m^*$ for $n\ge 3$ (p.196 in [Ad-Mi]).         







\section{extraspecial $p$-groups}



We assume that $p$ is an odd prime. 
The extraspecial $p$-group $E_n=p_+^{1+2n}$ is the group such that exponent is $p$,
its center is $C\cong \bZ/p$ and there is the extension
\[  \qquad 
\begin{CD}
0 @>>> C@>i>> E_n @>{\pi}>> V @>>> 0 \\
\end{CD}
\]
\noindent  with   $V=\oplus ^{2n}\bZ/p$. 
(For details of the cohomology of $E_n$ see [Te-Ya].) 
We can take generators  $a_1,...,a_{2n},c\in E_n$ such that $\pi(a_1),..
, \pi(a_{2n})$ (resp. $c$ ) make a base of $V$ (resp. $C$) such that
\[  \quad [a_{2i-1},a_{2i}]=c  \quad and \quad [a_{2i-1},a_j]=1 
\ \ if\ j\not = 2i. \]

We note that $E_n$ is also the central product of the $n$-copies of $E_1$
\[E_n\cong E_1\cdot \cdot \cdot E_1=E_1\times_{\la c\ra}E_1...\times_{\la c\ra} E_1.\]

Take cohomologies
\[
H^*(BC;\bZ/p)\cong \bZ/p[u]\otimes \Lambda(z), \quad   \beta z=u,
\]
\[
H^*(BV;\bZ/p)\cong \bZ/p[y_1,...,y_{2n}]\otimes \Lambda (x_1,...x_{2n}),
  \quad \beta x_i=y_i,\]
identifying the dual of $a_i$ (resp.$c$) with $x_i$ (resp. 
$z$).  That means
\[ H^1(E_n;\bZ/p)\cong Hom(E_n;\bZ/p)\ni x_i :a_j\mapsto \delta_{ij}.\]


The central extension  is expressed by 
\[
f=\sum _{i=1}^nx_{2i-1}x_{2i}\in H^2(BV;\bZ/p).
\]
Hence  $\pi ^*f=0$ in $H^2(BE_n;\bZ/p)$.
We consider the Hochshild-Serre spectral sequence
\[E_2^{*,*'}\cong H^*(BV;\bZ/p)\otimes H^*(BC;\bZ/p)\Longrightarrow
H^*(BE_n;\bZ/p).\]
Hence the first nonzero differential is $d_2(z)=f$ and the next differential is
\[ d_3(u)=d_3(Q_0(z))=Q_0(f)
 =\sum y_{2i-1}x_{2i}-y_{2i}x_{2i-1}.\]
In particular 
\[E_4^{0,*}\cong \bZ/p[y_1,...,y_{2n}]\otimes \Lambda (x_1,...x_{2n})/(f,Q_0(f)).\]


%Hence we know 
%\[w(j)=Q_jQ_0(f)=\sum_{i=1}^n y_{2i-1}^{p^j}y_{2i}-y_{2i-1}y^{p^j}_{2i}\]
%is also zero in $H^*(BE_n;\bZ/p)$.  The one of main theorems in [Te-Ya] is
%\begin{thm}([Te-Ya])
%The sequence $(w(1),...,w(n))$ is regular in $S_{2n}=\bZ/p[y_1,...,y_{2n}]$
%and
%\[S_{2n}/(w(1),....,w(n))\subset H^*(BE_n;\bZ/p).\]
%\end{thm}

\begin{lemma}  We have the inclusion
\[\Lambda(x_1,...,x_{2n})/(f)\subset H^*(BE_n;\bZ/p).\]
\end{lemma}
\begin{proof} We consider similar group $E_n'$ such that
its center is $C\cong \bZ/p$ and there is the extension
\[ 
\begin{CD}
0 @>>> C@>i>> E_n' @>{\pi}>> V' @>>> 0 \\
\end{CD}\]
but   $V'=\oplus ^{2n}\bZ_p$ such that there is the quotient map $q:E_n'\to E_n$.
We also consider the spectral sequence
\[E_2^{*,*'}\cong H^*(BV';\bZ/p)\otimes H^*(BC;\bZ/p)\Longrightarrow
H^*(BE_n';\bZ/p).\] 
Here $H^*(BV';\bZ/p)\cong \Lambda(x_1,...x_{2n})$. The first nonzero differential
is $d_2(z)=f$ but the second differential is
\[d_3(u)=\sum y_{2i-1}x_{2i}-y_{2i}x_{2i-1}=0.\]
Hence we have
\[H^*(BE_n';\bZ/p)\cong \bZ/p[u]\otimes\Lambda(x_1,...,x_{2n})/(f).\]
From the map $q^*: H^*(BE_n;\bZ/p)\to H^*(BE_n';\bZ/p)$,
we get the result.
\end{proof}
However $H^*(BE_n;\bZ/p)/(Ng)\not \cong  \Lambda(x_1,...,x_{2n})/(f)$, infact,
when $n=1$, from Theorem 3.3 in [Ya3]
we see
\begin{prop}
\[H^*(Bp_+^{1+2} ;\bZ/p)/(Ng)\cong \bZ/p\{1,x_1,x_2,a_1',a_2'\} \quad deg(a_i')=2.\]
\end{prop}

When $p=2$, the situation becomes   well.
The extraspecial $2$-group $D(n)=2_+^{1+2n}$ in the $n$-th central extension of the 
dihedral group $D_8$ of order $8$.  It has the central extension
\[ 0\to \bZ/2\to D(n)\to V\to 0\]
with $V=\oplus ^{2n}\bZ/2$.  Hence $H^*(BV;\bZ/2)\cong \bZ/2[x_1,...,x_{2n}]$.
Then using the Hochschild-Serre spectral sequence,  Quillen  proved [Qu]
\[H^*(BD(n);\bZ/2)\cong \bZ/2[x_1,...,x_{2n}]/(f,Q_0(f),...,Q_{n-2}(f))\otimes \bZ[w_{2^n}(\Delta)].\]
(In fact when $n$ is the real case (i.e., $n=-1,0,1\ mod(8)$), the cohomology
$H^*(BSpin_n;\bZ/2)$ injects into $H^*(BD(n);\bZ/2)$.)
Here $w_i$ (resp. $w_{2^n}(\Delta)$) is the Stiefel-Whitney class of usual representation from the above 
extension (resp. $2^n$-dimensional representation which restrict nonzero on the center).
Moreover Quillen proves following two theorems (Theorem 5.10-11 in [Qu])
\begin{thm} ([Qu]) $H^*(BD(n);\bZ/2)$ is detected by the product of cohomoloogy of maximal elementary abeian groups.
\end{thm}
\begin{thm} ([Qu]) The nonzero Stiefel-Whitney $w_i(\Delta)$ are those 
of degrees $2^n$ and $2^n-2^i$ for $0\le i<n$. 
\end{thm}
In fact $w_i(\Delta)$ generates the Dickson algebra 
in the cohomology of the maximal elementary abelian $2$-group
as stated in the preceding section. 
\begin{prop} When $n>2$, there is the surjection  
\[ \Lambda(x_1,...,x_{2n})/(f)\to H^*(BD(n);\bZ/2)/(Ng).\]
\end{prop}
\begin{proof}
By the same arguments with $p=odd$, we see
\[\Lambda=\Lambda(x_1,...,x_{2n})/(f)\subset H^*(BD(n);\bZ/2).\]
The fact $w_2(\Delta)=0$ follows from the above second Quillen's theorem.
Hence we have $w_{2^n}(\Delta)\in Ng$ from Becher's theorem (Theorem 6.2).
Thus we get the theorem.
\end{proof}


Let $0\not =x\in \Lambda$.  Then by the Quillen's theorem, $i_A^*(x)\not =0$ for
\[ i_A^*: H^*(BD(n);\bZ/2)\to H^*(BA;\bZ/2)\]
with $A\cong \bZ/2\oplus...\oplus \bZ/2$.  Let $H^*(BA;\bZ/2)/(Ng)\cong \Lambda(x_1',...,x_m')$.
Then $i_A^*(x_i)\in \bZ/2\{x_1',...,x_m'\}$  and hence the map is restricted
\[ i_A^* : \Lambda(x_1,...,x_{2n})/(f) \to \Lambda(x_1',...,x_m').\]
However this map is not need injective.  In fact there is a possibility
of $i_A^*(x)\in Ng(A)$, e.g., $i_A^*(x_1x_2)=(x_1')^2\in Ng(A)$. 





\section{unramified theory}


In this section, we assume that
$k$ is an $algebraic$ $closed$ $field$ of $ch(k)=0$.

  
Let $K$ be a function field of $k$, that is finitely generated as a field over $k$.
Here we recall the definition of unramifed cohomology of $H^*(K;\bZ/p)$ according to 
 Saltman, Peyre and Colliot-Thelene. 
We denote by $P(K/k)$ the set of discrete valuation rings $A$ of rank one
such that $k\subset A\subset K$ and that the fraction field $Fr(A)$ of $A$ is $K$.
If $A$ belongs to $P(K/k)$, then for the residue field $\kappa _A$, we can define
the residue map
$\partial_A:H^*(K;\bZ/p)\to H^{*-1}(\kappa_{A};\bZ/p)$ as follows.

Let $\hat K_A$ be the completion, $\hat K_A^{nr}$ the maximal unramified extension of $K_A$
and $\bar K_A$ is an algebraic closure of $K_A$.
 Put $I_A=Gal(\bar K_A/\hat K_A^{nr})$ and $G_A=Gal(\bar K_A/\hat K_A)$.
\[\bar K_A\stackrel{I_A}{--}   \hat K_A\stackrel{G_A/I_A}{--}  \hat  K_A\stackrel{}{--} K
   .\]
 Then $\partial_A$ is defined as the composition of maps
\[\partial : H^*(K;\bZ/p)\to H^*(\hat K_A;\bZ/p)\]
\[\stackrel{proj.}{\to} H^{*-1}(G_A/I_A)\otimes H^1(I_A;\bZ/p)\cong H^{*-1}(\kappa_A;\bZ/p).\]
Here we used that $\hat K_A\cong I_A\oplus (G_A/I_A)$ ( [Sa]) and $I_A\cong \hat \bZ$.
Moreover
$H^*(\hat \bZ;\bZ/p)\cong \bZ/p$ if $*=0,1$ and $\cong 0$ otherwise.


Then we can define  the unramified cohomology
\[H_{nr}^*(K;\bZ/p)=\cap_{A\in P(K/k)} Ker(H^*(K;\bZ/p)\stackrel{\partial_A}{\to}H^{*-1}(\kappa_A;\bZ/p)).\]

Namely, when $X$ is complete, the residue map is the same as the differential $d_1$ of the
coniveau spectral sequence given in $\S 2$, and hence the unramified cohomology is just 
the $E_2$-term $H^0_{Zar}(X;H_{\bZ/p}^m)$ of the coniveau spectral sequence.

\begin{cor}
When $X$ is complete, there is the isomorphism
\[H_{nr}^*(k(X);\bZ/p)\cong H^{*,*}(X;\bZ/p)/(\tau)\oplus 
Ker(\tau)|H^{*+1,*-1}(X;\bZ/p).\]
\end{cor}

Now we consider the case $X=BG$ ; non complete cases.
Let $W//G$ be the scheme which has the coordinate  ring $k[W]^G$.  Then it is known that
$W//G$ contains $U_n/G$ as an open set.
Here $U_n$ is the open set given in $\S 3$, where $G$ acts freely. 
Hence we have for $*<n$
\[H^0_{Zar}(W//G;H_{\bZ/p}^*)\subset H^{0}_{Zar}(U_n/G;H^*_{\bZ/p}).\]
Since $G$ is  reductive, it is also known that
the fraction field of $k[W]^G$ is $k(W)^G$, that is $k(W//G)=k(W)^G$.
Thus we get
\begin{lemma}  For $*<n$, we have
\[H_{nr}^*(K(W)^G;\bZ/p)=H_{Zar}^0(W//G;H^*_{\bZ/p})\qquad \qquad\]
\[\qquad \qquad \subset H^0(U_n/G;H^*_{\bZ/p})=H^0(BG;H_{\bZ/p)}^*).\]
\end{lemma} 



According to Peyre [Pe], we define a subring $H_{nr}^*(G;\bZ/p)$ of $H^*_{et}(BG;\bZ/p)$
as follows.
Let $P(G)$ be the set of elements $g\in G$ such that
$I=\la g\ra\cong \bZ/p^s$ for some $s\ge 1$ but $g\not =h^p$ for any $h\in G$.
Then the centralizer is written as
\[Z_G(I)\cong I\oplus H\qquad with \ H=Z_G(I)/I.\]
Let us write  by  $\partial _g$  the composition map 
\[ \partial_g : H^*(BG;\bZ/p)\to H^*(BZ_{G}(I);\bZ/p)\]
\[ \cong H^{*-*'}(BH;\bZ/p)\otimes H^{*'}(BI;\bZ/p)
\stackrel{proj.}{\to} H^{*-1}(BH;\bZ/p)\]
using $H^1(BI;\bZ/p)\cong \bZ/p$.  Then define the unramified cohomology by
\[H_{nr}^*(G;\bZ/p)=\bigcap _{g\in P(G)}Ker(H^*(BG;\bZ/p)\stackrel{\partial_{g}}{\to}
                           H^{*-1}(BH;\bZ/p)).\]
{\bf Remark.} The restriction map 
$H^1(B\la g\ra;\bZ/p)\to H^1(B\la g^p\ra;\bZ/p)$ is always zero. Hence we need not 
consider the case $I=\la g^p\ra$.                             
                           

  
 \begin{thm}(Peyre [Pe])                    
 Let $W$ be a faithful representaion of $G$.  Let $q$ is the quotient map
 $q: Gal(\bar{k(W)}/k(W)^G) \to Gal(k(W)/k(W)^G)=G.$
 Then
\[q^*(H_{nr}^*(G;\bZ/p))\subset H_{nr}^*(k(W)^G ;\bZ/p).\]
\end{thm}
\begin{proof}
 Arguments of pages 203 to 206 in the proof of Proposition 3 in [Pe]
work exchanging $H^3(-;\bQ/\bZ(-))$ to $H^*(-;\bZ/p)$.
Indeed we have the commutative diagram ((12) in [Pe])
\[\begin{CD}
H^*(G;\bZ/p)@>{\partial_{D,g}}>> H^{*-1}(D;\bZ/p)\\
     @VVV    @VVV    \\
H^*(k(W)^G;\bZ/p)  @>{\partial_A}>> H^{*-1}(\kappa_A;\bZ/p)
\end{CD} \]
where $D$ is the decomposition subgroup of $G$.
If $x\in H_{nr}^*(G;\bZ/p)$,  then $\partial_{D,g}(x)=0$
and hence $\partial_A(\rho^*(x))=0$.
\end{proof} 
It is well known (Theorem 4.1.5 in [Co-Te]) that if $K$ is purely
transcendental over $k$ (i.e., $K\cong k(x_1,...,x_n)$ for indeterminate $x_i$),
then
\[H^*(k;\bZ/p)\cong H_{nr}^*(K;\bZ/p)\quad for \ *>0.\]
When $k$ is algebraic closed field, it is immediate $H^+(k;\bZ/p)=0$.

\begin{cor}
Suppose that
\[0\not= x\in H^{*}(BG;\bZ/p)/(Ng) \quad and \quad x\in H_{nr}^*(G;\bZ/p).\]
Then $q^*(x)\not= 0$ in $H_{nr}^*(k(W)^G;\bZ/p)$.  Hence $k(W)^G/k$
is not purely trascendental.
\end{cor}
\begin{proof}
From the preceding theorem, we have 
\[q^*(x)\in H_{nr}^*(k(W)^G;\bZ/p)\subset H^*(k(W)^G;\bZ/p).\]
It is nonzero since 
\[ H^*(BG;\bZ/p)/(Ng)\subset H^0_{Zar}(BG;H_{\bZ/p}^*)\subset H^*(k(W)^G;\bZ/p).\]
\end{proof}
\begin{cor}
Let $G$ be a $p-group$ of exponent $p$. If $H^2(BG;\bZ/p)/(Ng)$
 is not detected by $\bZ/p\times \bZ/p$, then
$k(W)^G$ is not purely transcendental.
\end{cor}  
\begin{proof}
Suppose that $x\not \in H_{un}^2(G;\bZ/p)$.  Then we can take $x= \sum x_1x_2$ such that
$x_1|H^1(BI;\bZ/p)\not =0$ with $I\cong \bZ/p$ and  for $H=Z_G(I)/I$,
$x_2|H^1(BH;\bZ/p)\not =0$. Here
\[H^1(BH;\bZ/p)/(Ng)=H^1(BH;\bZ/p)\cong Hom(H,\bZ/p).\]
 Hence $x_2$ defines
subgroup $J\cong \bZ/p$ of $H$ such that $x_2|H^1(BJ;\bZ/p)\not =0$
(since $H$ is exponent $p$).  Thus
the element $x$ is detected by the subgroup $I\oplus J\cong \bZ/p\oplus \bZ/p$.
\end{proof}




     
       
       
       
       
       
       
       
              
  \section{ Saltman's example}
  
  
  
Let $G$ be the group defined by
\[0\to \la c_3,c_4\ra \to G \to E_2=p_+^{1+4}\to 0\]
\[ with \quad [a_1,a_3]=c_3, \quad and \quad [a_1,a_4]=c_4.\]
Saltman and Bogomolov showed that $H^2_{nr}(k(W)^G;\bQ/\bZ)\not =0$
for this group.  We will see the $\bZ/p$-coefficient case. 
\begin{lemma} The $3$-dimensional cohomology $H^3(BG;\bZ/p)$ contains
the $\bZ/p$-module
\[A=\bZ/p\{y_ix_j|1\le i,j\le4\}/(Q_0(f),Q_0(x_1x_3),Q_0(x_1x_4)).\]
\end{lemma}
\begin{proof}
Consider the central extension
\[ 0\to \la c,c_3,c_4\ra \to G\to V=\oplus ^4\bZ/p\to 0\]
and induced spectral sequence
\[E_2^{*,*'}\cong \bZ/p[y_1,...,y_4,u,u_3,u_4]\otimes \Lambda(x_1,...,x_4,z,z_3,z_4)\]
converging to $H^*(BG;\bZ/p)$.  The first differential is
\[d_2(z)=f,\ \  d_2(z_3)=x_1x_3,\ \ d_2(z_4)=x_1x_4.\]
The second nonzero differential is
\[ d_3(u)=Q_0(f),\ \ d_3(u_3)=Q_0(x_1x_3),\ \ d_3(u_4)=Q_0(x_1x_4).\]
Thus we see
\[ A\cong E_4^{3,0}\cong  E_{\infty}^{3,0}\subset H^3(BG;\bZ/p).\]
\end{proof}  


\begin{thm} ( Saltman [Sa])  We have
\[ 0\not =x_1x_2\in H^2_{nr}(G;\bZ/p)\cap H^{2,2}(BG;\bZ/p)/(\tau).\]
Hence $k(W)^G$ is not purely transcendental.
\end{thm}
\begin{proof} 
From the preceding lemma, we see $Q_0(x_1x_2)\not =0$ in $H^3(BG;\bZ/p)$.
Hence  we see $x_1x_2\not \in Ng(G)$ from Lemma 3.4.

Next we will show $x_1x_2\in H_{nr}(G;\bZ/p)$.
Suppose this is not the case.  By the definition, this means that
there is an element $g$ and $h\in Z_{G}(\la g\ra)$ such that
\[ x_i|\la g\ra \not =0,\quad x_j|\la h\ra\not =0\quad for\ \{i,j\}=\{1,2\}.\]

Let us write
\[a^{\Lambda}=a_1^{\lambda_1}...a_4^{\lambda_4}\quad for\ \Lambda=(\lambda_1,...,\lambda_4),\]
\[a^{M}=a_1^{\mu}...a_4^{\mu_4}\quad for\ M=(\mu_1,...,\mu_4).\]
Note that $x_i|\la a^{\Lambda}\ra=\lambda_i$.
The commutator is given by the definition
\[[a^{\Lambda},a^M]=c^dc_3^{d_3}c_4^{d_4}
\quad where\ \ d=\lambda_1\mu_2-\lambda_2\mu_1+\lambda_3\mu_4-\lambda_4\mu_3,\]
\[ d_3=\lambda_1\mu_3-\lambda_1\mu_3,\qquad   d_4=\lambda_1\mu_4-\lambda_4\mu_1.\]

Take $g=a^{\Lambda}$.  Exchanging $x_1\mapsto x_1+\lambda x_2$ or $x_1\mapsto x_2$
(if necessary),
we can take 
\[\lambda_1=1,\lambda_2=0\quad \Lambda=(1,0,\lambda_3,\lambda_4).\]
 (Hence $x_1\la g\ra=1$ and $x_2|\la g\ra=0$.)
Then we can take $h=a^M$ so that $x_2|\la h\ra=1$ and $h\not \in \la g\ra$, that means
\[\mu_1=0,\ \mu_2=1\quad M=(0,1,\mu_3,\mu_4).\]

From $d_3=0$ and $d_4=0$, we see 
\[\mu_3=\lambda_3\mu_1=0\quad and\quad \mu_4=\lambda_4\mu_1=0\]
(and hence $M=(0,1,0,0)$).  Therefore $d=1\times 1-0+0-0=1$.  This is a contradiction. 
Hence we have proved  $x_1x_2 \in H_{nr}(G;\bZ/p)$.
\end{proof}

\section{non detected examples}

%We consider the other groups.
%Let $G_m$ be the central product of $m$-th copies of $G$
%\[ G_m=G\cdot...\cdot G=G\times _{\la c\ra}G\times_{\la c\ra}... \times_{\la c\ra}G\]
%which is generated by $a_1,...,a_{4m}$, $c$, $c_3,c_4,c_7,...,c_{4m-1},c_{4m}$.
%Of course there is the central extension
%\[0\to \la c_3,c_4,...,c_{4m-1},c_{4m}\ra \to G_m\to E_{2m}=p_+^{1+4m}\to 1.\]
%The comutators are given for $1\le i\le m$
%\[ [a_{4i-3},a_{4i-2}]=[a_{4i-1},a_{4i}]=c,\]
%\[ [a_{4i-3},a_{4i-1}]=c_{4i-1},\quad and \quad [a_{4i-3},a_{4i}]=c_{4i}.\]
%
%Let us write
%$a^{\Lambda}=a_1^{\lambda_1}...a_{4m}^{\lambda_{4m}}$ and $\Lambda=(\lambda_1,...,\lambda_{4m})$
%as the case $m=1$.  Similarly we define $a^M$.
%Note also $x_i|\la a^{\Lambda}\ra=\lambda_i$.
%The commutator is given by the definition
%\[[a^{\Lambda},a^M]=c^dc_3^{d_3}c_4^{d_4}....c_{4m}^{d_{4m}}\]
%\[ where\ \ d=\sum _i\lambda_{4i-3}\mu_{4i-2}-\lambda_{4i-2}\mu_{4i-3}
%+\lambda_{4i-1}\mu_{4i}-\lambda_{4i}\mu_{4i-1},\]
%\[ d_{4i-1}=\lambda_{4i-3}\mu_{4i-1}-\lambda_{4i-1}\mu_{4i-3},\qquad  
% d_{4i}=\lambda_{4i-3}\mu_{4i}-\lambda_{4i}\mu_{4i-3}.\]
% 
% \begin{lemma}
% Let $\xi_m=x_1x_2x_5x_6....x_{4m-3}x_{4m-2}$.  Then  $\xi\in H_{nr}(G_n;\bZ/p)$.
% \end{lemma}
% \begin{proof}
% Suppose that $\xi_m\not \in H_{un}^*(G_m;\bZ/p)$.
% This means there is a series of elements $g=g_1, g_2,...,g_{2m}$ such that
% $g_i\in Z_G(\la g\ra)$ and
% \[ \xi_m|\la g_1,...,g_{2m}\ra\not =0,\quad and \quad x_i|\la g\ra\not =0\ \ for \ some \ i.\]
% 
% Take generators adequately, let $g=a^{\Lambda}$ with $\lambda_1=1$ and $\lambda_2=0$.
% Then we can take $g_i=a^{\Lambda_i}$ such that
% \[ \Lambda_1=(1,0,*,*|*,*,*,*|....|*,*,*,*),\]
% \[ \Lambda_2=(0,1,*,*|*,*,*,*|....|*,*,*,*),\]
%\[ \Lambda_5=(0,0,*,*|1,*,*,*|....|*,*,*,*),\]
%  \qquad \qquad    .......
% \[ \Lambda_{4m-3}=(0,0,*,*|0,0,*,*|....|1,*,*,*),\]
% \[ \Lambda_{4m-2}=(0,0,*,*|0,0,*,*|....|0,1,*,*).\]
% Moreover taking off $\Lambda_i,i\ge 3$ (if necessary), we can chage $\Lambda_2$ so that
%  \[ M=\Lambda_2'=(0,1,0,0|0,0,*,*|....|0,0,*,*),\]
%  where $\mu_3=\mu_4=0$ follows from $d_3=d_4=0$ as the case $m=1$.
%  
%  Let $i\ge 2$.  If $\lambda_{4i-3}\not=0$, then $d_{4i-1}=0$ implies
%  \[ \mu_{4i-1}=(\lambda_{4i-1}/\lambda_{4i-3})\mu_{4i-3}=0\quad (since\ \mu_{4i-3}=0),\]
%  and similarly $\mu_{4i}=0$.  If $\lambda_{4i-3}=0$,  then  we see
% $\lambda_{4i-1}=\lambda_{4i}=0$ ; because there is $\Lambda_{j}=M'$ such that
% $\mu _{4i-3}'=1$ and 
% \[ \lambda_{4i-1}=(\mu_{4i-1}'/\mu_{4i-3}')\lambda_{4i-3}=0.\]
% Hence any cases, we see 
% \[\lambda_{4i-3}\mu_{4i-2}-\lambda_{4i-2}\mu_{4i-3}
%+\lambda_{4i-1}\mu_{4i}-\lambda_{4i}\mu_{4i-1}=0.\]
%Then $d=1+0+...+0\not =0$.  This is a contradiction.
%\end{proof}

%Unfortunately, we can not prove $0\not =\xi \in H^*(BG_m;\bZ/p)/(Ng)$.
We consider  group $G_m'$ and element $\xi_m\in H^*(BG_m';\bZ/p)/(Ng)$
which is not detected  by elementary abelina $p$-groups, while
it  is not in $H_{nr}(G_m';\bZ/p)$.


Let $G'$ be the group defined by
\[0\to \la c_3,c_4\ra \to G' \to E_2=p_+^{1+4}\to 0\]
\[ with \quad [a_1,a_3]=c_3, \quad and \quad [a_2,a_4]=c_4.\]
Difference of the definitions of $G$ and $G'$ is just $[a_2,a_4]=c_4$.
However note $G'$ has the rank $2$ elementary abeilan $p$-subgroup
\[ \la a_1a_3^{-1},a_2a_{4}^{-1}\ra \cong \bZ/p\oplus \bZ/p\]
while $G$ does not.

Let $G_m'$ be the central product of $m-1$-th copies of $G'$
and one $G$
\[ G_m'=G\cdot G'\cdot...\cdot G'=G\times_{\la c\ra}G'\times _{\la c\ra}... \times_{\la c\ra}G'\]
which is generated by $a_1,...,a_{4m}$, $c$, $c_3,c_4,c_7,...,c_{4m-1},c_{4m}$.
%Of course there is the central extension
%\[0\to \la c_3,c_4,...,c_{4m-1},c_{4m}\ra \to G_m\to E_{2m}=p_+^{1+4m}\to 1.\]
The comutators are given for $1\le i\le m$
\[ [a_{4i-3},a_{4i-2}]=[a_{4i-1},a_{4i}]=c,\]
\[ [a_{4i-3},a_{4i-1}]=c_{4i-1},\quad and \quad [a_{4i-2},a_{4i}]=c_{4i}\ for\ i\not =1,\]
\[ but \quad [a_1,a_4]=c_4.\]

Let us write
$a^{\Lambda}=a_1^{\lambda_1}...a_{4m}^{\lambda_{4m}}$ and $\Lambda=(\lambda_1,...,\lambda_{4m})$.
Note also $x_i|\la a^{\Lambda}\ra=\lambda_i$.
The commutator is given by the definition
\[[a^{\Lambda},a^M]=c^dc_3^{d_3}c_4^{d_4}....c_{4m}^{d_{4m}}\]
\[ where\ \ d=\sum _i\lambda_{4i-3}\mu_{4i-2}-\lambda_{4i-2}\mu_{4i-3}
+\lambda_{4i-1}\mu_{4i}-\lambda_{4i}\mu_{4i-1},\]
\[ d_{4i-1}=\lambda_{4i-3}\mu_{4i-1}-\lambda_{4i-1}\mu_{4i-3},\qquad  
 d_{4i}=\lambda_{4i-2}\mu_{4i}-\lambda_{4i}\mu_{4i-2}\ (i\not =1),\]
 \[ and \qquad  d_{4}=\lambda_{1}\mu_{4}-\lambda_{4}\mu_{1}.\] 


 \begin{lemma}
 Let $\xi_m=x_1x_2x_5x_6....x_{4m-3}x_{4m-2}\in H^*(BG_m';\bZ/p)$.  Then  $\xi_m\in H^*(BG_n';\bZ/p)/(Ng)$ 
 is not detected by elementary abelian $p-groups$.
 \end{lemma}
 \begin{proof}
 Suppose that $\xi_m$ is detected by 
\[\la g_1, g_2,...,g_{2m}\ra \cong \bZ/p\oplus ...\oplus \bZ/p\]
 such that
 $\xi_m|\la g_1,...,g_{2m}\ra\not =0$, and $x_i|\la g_i\ra\not =0$ for some $i$.
  Take generators adequately, let  $g_i=a^{\Lambda_i}$ such that
 \[ \Lambda_1=(1,0,*,*|0,0,*,*|....|0,0,*,*),\]
 \[ \Lambda_2=(0,1,*,*|0,0,*,*|....|0,0,*,*),\]
\[ \Lambda_5=(0,0,*,*|1,0,*,*|....|0,0,*,*),\]
\[ \Lambda_6=(0,0,*,*|0,1,*,*|....|0,0,*,*),\]
  \qquad \qquad    .......
 \[ \Lambda_{4m-3}=(0,0,*,*|0,0,*,*|....|1,0,*,*),\]
 \[ \Lambda_{4m-2}=(0,0,*,*|0,0,*,*|....|0,1,*,*).\]
 
 Let $i>1$. Consider the commutativity of $L=\Lambda_{4i-3}$ and $M=\Lambda_1$.
 Since $d_{4i-1}=\lambda_{4i-3}\mu_{4i-1}-\lambda_{4i-1}\mu_{4i-3}=0$, we see
 \[\mu_{4i-1}=0\quad from\ \mu_{4i-3}=0 \ and \ \lambda_{4i-3}=1.\]
 Similarly we have $\mu_{4i}=0$ from the commutativity of $\Lambda_{4i-2}$ and $\Lambda_1$.
 Thus we see
\[ \Lambda_1=(1,0,*,*|0,0,0,0|....|0,0,0,0).\]

We also have $\Lambda_2=(0,1,*,*|0,0,0,0|....|0,0,0,0).$
Let $\Lambda=\Lambda_1$ and $M=\Lambda_2$.  By the commutativity and facts
$d_3=d_4=0$, we see $\mu_3=\mu_4=0$, that is, 
\[ \Lambda_2=(0,1,0,0|0,0,0,0|....|0,0,0,0).\]
However this is a contradiction, indeed,
\[d=\sum _i\lambda_{4i-3}\mu_{4i-2}-\lambda_{4i-2}\mu_{4i-3}
+\lambda_{4i-1}\mu_{4i}-\lambda_{4i}\mu_{4i-1}=1\not =0.\]
\end{proof}


  
\begin{lemma}
 $ Q_0...Q_{2m-2}(\xi_m)\not =0$ in $H^*(BG_m;\bZ/p).$
 \end{lemma}
 \begin{proof}
 We recall that $G_m'$ is the central product of $G$ and copies of $G'$,  that means
 \[ 0\to \oplus^m\bZ/p \to G\times G'\times ...\times G' \stackrel{pr.}{\to} G_m'\to 0.\]
 The map induces the map of cohomologies
 \[pr.^*:  H^*(BG_m';\bZ/p)\to H^*(BG;\bZ/p)\otimes H^*(BG';\bZ/p)^{\otimes n}.\]
 
 The operation $Q_i$ is derivative, we have
 \[Q_{0}...Q_{2m-2}(\xi_m)=\]
  \[Q_{0}(x_1x_2)Q_1Q_2(x_5x_6)...Q_{2m-3}Q_{2m-2}(x_{4m-3}x_{4m-2})+.....\]
  Here from Lemma 12.1,
  \[Q_1(x_1x_2)=y_1x_2-y_2x_1\not =0\quad in\ H^*(BG;\bZ/p).\]
For $i>1$, we see
  \[ Q_{2i-3}Q_{2i-2}(x_{4i-3}x_{4i-2})
  =y_{4i-3}^{p^{2i-2}}y_{4i-2}^{p^{2i-3}}-y_{4i-3}^{p^{2i-3}}y_{4i-2}^{p^{2i-2}}.\]
  This is nonzero in $H^*(BG';\bZ/p)$ because  it is nonzero in the restriction image 
  \[ H^*(BG';\bZ/p)\to H^*(B\la a_1a_3^{-1},a_2a_{4}^{-1}\ra;\bZ/p)\cong \bZ/p[y_1',y_2']\otimes 
\Lambda(x_1',x_2')\]
where  $y_{4i-3},y_{4i-1}\mapsto y_1'$ and $y_{4i-2},-y_{4i}\mapsto y_2'.$
       \end{proof}
       \begin{cor}  Elements in $H^{2m}(BG_m';\bZ/p)/(Ng)$ is not detected by abelian subgroups, 
namely, 
    the restriction map
       \[ Res : H^{2m}(BG_m';\bZ/p)/(Ng)\to \Pi_{A;abelian}H^{2m}(BA;\bZ/p)/(Ng)\]
       is not injective.
       \end{cor}
       



 
 
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\newblock  Coniveau spectral sequence and motivic cohomology of quadrics.
\newblock \emph{submitting.} (2005).


\bibitem[Ya3]{Ya3}
N.~Yagita.
\newblock  Coniveau filtration of cohomology of groups.
\newblock \emph{submitting.} (2008).




































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